For Monday at 9 AM, please write a 1-2 paragraph personal reflection on your blog that follows up on our discussion/ debate today on instrumental and relational learning. Your reflection should consider ways that "fluency" and "meaning-making" might be integrated in math teaching and learning. Please give an example related to a particular math topic if possible.
Also, please take a look at the blog postings about Assignment #1, and think about what topic(s) you are interested in and who you might like to work with in your group. If you have requests and suggestions about choice of group or topic, feel free to email them to me -- but we will not settle on anything till the end of our next class, where we will discuss this. I may also add some other possible project topics before Monday!
For the whole weekend, I have been thinking about the relation between relational understanding and instrumental understanding in mathematics. As defined in the previous article, instrumental understanding focuses on how to use methods/algorithms to solve problems, while relational understanding concentrates more on why the method can solve the problem.
It is widely accepted that both of them are important in mathematics teaching. Even though in class we stand in two different groups and seemingly argued against the other side, we all believed that both of the two understandings play crucial roles in mathematics.
However, we do have different opinions toward what kind of understanding should be put more effort in class. My understanding of this question, after the whole weekend thinking and discussion, is it depends.
My logic stands in the following way. Consider our students in three different levels: (a) those who are very good at and interested in mathematics, whose scores are A level or above; (b) those who have trouble understanding the internal reasons behind conclusions, whose scores are below C level; and (c) those who are in the middle, whose scores are in the between of C level and B level.
For type (a) students, both instrumental and reasonal understands are important, as reasonal understanding provides them interests of learning mathematics, and instrumental understanding helps them solve problems efficiently. I firmly believe the two understandings form a spirical process that carries students interests and understandings of mathematics forward. Meanwhile, with or without us, these students can find some way learning and understanding mathematics. In short, they are not the students worths of discussion in a "normal" mathematics class.
For type (b) students, it is certain that forcing them to memorize some process of solving problems can be a short-run but not a long-run method of raising their math scores.There are so many "outsider" factors influencing the results of their mathematical learning (for example, personal health problems, social-economic problems, etc.). The solution should be a mixed bundle including but not limited with the two types of understandings.
If we assume type (c) students are the majority of a "normal" mathematics class, then instrumental understanding should be an efficient way of showing students how mathematics are applied in a problem-solving process. The reality is that if a teacher spends too much time on reasoning why a mathod works (relational understanding), since most of these students need time to "catch and feel" the reasoning process----and in this case they are hard to understand what the teacher says in such a short time----they will quickly feel bored and give up. My cousin (Grade 7) mentioned an interesting story in his math class:
"My math teacher is some how boring but entertaining. He spent 80 minutes explaining 4 questions that I can solve in 3 minutes, because most of the people in my class have not yet learned about this kind of stuff. He explained the same stuff over and over again, but always add some little examples that make us crack up." (Here is the link to his blog: http://minerbill1234.blogspot.ca/2015/09/2015-09-15.html)
And thus I believe there should not be a standard answer to the question "which is more important in mathematical learning, instrumental understanding or relational understanding". It is totally depended on students' situations and teachers' teaching skills. I personally believe 20% of reasoning and 80% of practising is a possible bundle to start with.
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